Voter Turnout: Theory and Evidence
from Texas Liquor Referenda∗
Michael Conlin
Department of Economics
Syracuse University
Syracuse, NY 13244
meconlin@maxwell.syr.edu
Stephen Coate
Department of Economics
Cornell University
Ithaca, NY 14853
sc163@cornell.edu
Revised October 2002
Abstract
This paper uses data from Texas liquor referenda to explore a new approach to understanding voter turnout, inspired by the theoretical work of
Harsanyi (1980) and Feddersen and Sandroni (2002). It presents a model
based on this approach and structurally estimates it using the referendum
data. It then compares the performance of the model with two alternative
models of turnout. The results are encouraging: the structural estimation yields reasonable parameter estimates and the model performs better
than the two alternatives considered.
∗ We thank Tim Besley, Mark Bils, Stacy Dickert-Conlin, Tim Feddersen, Liz Gerber, John
Matsusaka, Andrea Moro, Ted O’Donoghue, Jan Ondrich, Alvaro Sandroni, Tim Vogelsang
and seminar participants at Boston University, Brown, Columbia, Cornell, Rochester, Syracuse, USC, UC San Diego, and Yale for helpful comments and discussions. We are also
indebted to Randy Yarborough at the Texas Alcohol Beverage Commission for providing the
local liquor law election data.
1
1
Introduction
In Texas, as in most of the United States, the sale of liquor is heavily regulated.
Moreover, regulations vary across different localities within the state. Some
are “dry”, completing prohibiting the retail sale of alcoholic beverages. Others
permit the sale of only certain types of liquor or require that liquor be consumed
“off premise”. What is particularly interesting about Texas is that alcohol
regulations at the local level are determined directly by the citizens.1 A citizen
wishing to change regulations in his community can get the change voted on in a
referendum. Such referenda are commonplace, with over 500 elections between
1976 and 1996.
These liquor referenda appear a promising vehicle for understanding voter
turnout. First, turnout varies widely. In some communities over 75% of the
voting age population show up to vote, while in others turnout is less than 10%.
Second, since there is a limited set of regulations that are actually proposed,
the issues decided by the referenda are basically the same across jurisdictions.
Third, the referenda are typically held separately from other elections, so that
the only reason to go to the polls is to vote on the proposed change in liquor
law.
This paper uses data from these referenda to explore a new approach to understanding voter turnout, inspired by the theoretical work of Harsanyi (1980)
and Feddersen and Sandroni (2002). It presents a model based on this approach
and structurally estimates it using the referendum data. It then compares the
performance of the model with two alternative models of turnout. The results are encouraging: the structural estimation yields reasonable parameter
estimates and the model performs better than the two alternatives considered.
The approach begins with the observation that a referendum creates a contest between two groups: those who support it and those who oppose it. The
winner of the contest is the group the most of whose members vote. Individual
group members are presumed to want to “do their part” to help their side win.
This is not because they receive a transfer from other group members for doing
so - they simply adhere to the belief that this is how a citizen should behave
in a democracy. In the spirit of Harsanyi (1980) and Feddersen and Sandroni
(2002), “doing their part” is understood to mean following the voting rule that,
if followed by everyone else on their side, would maximize their side’s aggregate
utility. Thus, individuals are assumed to act as group rule-utilitarians, with
their “groups” being those who share their position.2
The optimal voting rule specifies a critical cost level below which an individual should vote. A lower critical level creates more turnout and hence raises the
probability of the group’s preferred outcome. On the other hand, it increases
1 Local jurisdictions do not have complete discretion over liquor regulation. For example,
the State of Texas prohibits the sale of alcoholic beverages in certain residential areas and
within 300 feet of a church, school or hospital. There are also statewide restrictions on the
times alcohol may be sold.
2 A rule-utilitarian follows the rule that if followed by all citizens would maximize aggregate
utility. By analogy, a group rule-utilitarian follows the rule that if followed by all group
members would maximize aggregate group utility.
2
the expected voting costs incurred by group members. Balancing these two considerations determines the optimal critical level. The optimal voting rule will
depend upon the expected turnout from the opposition. The higher this is the
larger the critical level must be in order to ensure any given chance of success.
Accordingly, it is natural to think of the voting rules as being determined in a
game in which individuals from the two groups move simultaneously. In equilibrium, individuals in each group must be satisfied with their voting rule, given
the rule they expect the opposing group to choose.
The equilibrium voting rules depend on election specific characteristics like
the relative sizes of the two competing groups, the importance of the issue
decided by the referendum to the groups, and expected voting costs. Understanding these relationships yields predictions for how turnout should depend
on election specific characteristics. In this way, the approach yields a theory of
turnout.
The specific model developed in the paper assumes that all supporters enjoy
the same benefit and all opposers incur the same cost if the referendum passes.
This sidesteps the interesting question of how the burden of voting should be
shared among group members with differing intensities of preference. In addition, the fraction of supporters is assumed to be the realization of a random
variable with a Beta distribution. Individuals do not observe the realization but
do know the parameters of the Beta distribution. This captures the idea that
individuals will be aware of general characteristics of their fellow citizens - such
as age and religious affiliation - that will influence the likely distribution of supporters. Finally, the cost of voting for each supporter and opposer is assumed
to be the realization of an independent random variable uniformly distributed
on an identical support. The model is therefore described by five parameters:
the benefit of the proposed change to supporters; the cost to opposers; the two
parameters of the Beta distribution; and the upper bound of the support of
voting costs. The equilibrium voting rules for supporters and opposers depend
on these parameters and these, together with the realization of the fraction of
supporters, determine the turnout of supporters and opposers.
Our data include information on the type of referendum, the votes for and
against, and when the referendum was voted on. We also know the size of the
voting population and many characteristics of the jurisdiction in question at
the time of the election. This includes the religious affiliations of the county
population and the liquor regulations in neighboring communities. Finally, we
know weather conditions on the day of voting. We specify functional forms which
relate our parameters to these characteristics and then estimate the coefficients.
In this procedure, we draw on the work of Shachar and Nalebuff (1999).
The two alternative models we consider maintain the same underlying assumptions concerning the environment but postulate different voting behavior.
They are ad hoc, in the sense that they do not provide an account of why people behave in the postulated way. Nonetheless, they do capture ideas that have
been expressed in the literature. The intensity hypothesis says that people are
more likely to vote the more intensely they feel about an issue. The popularity
hypothesis says that individuals are more likely to vote if they believe that many
3
of their fellow citizens share their position on the issue. Using the non-nested
hypothesis test of Vuong (1989), we reject the hypotheses that the intensity
and popularity models and our group rule-utilitarian model are equally close to
the true data generating process in favor of the alternative hypothesis that our
model is closer.
The organization of the remainder of the paper is as follows. The next section explains how the paper relates to previous work on voter turnout. Section
3 describes the institutional details concerning the referenda that we study and
presents the raw data. Section 4 presents our group rule-utilitarian model. Section 5 describes how we estimate it and section 6 discusses the results. Section
7 outlines the two alternative models and compares their performance with the
group rule-utilitarian model. Section 8 concludes with suggestions for further
research.
2
Relationship to the turnout literature
Understanding voter turnout is a central problem in political economy. Turnout
is sensitive to the specific characteristics of elections. Political parties understand this and policy stances are fashioned to “bring out the base” or discourage
the opposition’s base. Accordingly, turnout not only determines which option
wins but also shapes the policy options from which voters select. Reflecting this
importance, there has been a considerable amount of work on the subject. Here,
we briefly point out where our paper fits into the literature. The reader is referred to Aldrich (1993), (1997), Fiorina (1997), Grossman and Helpman (2001),
and Matsusaka and Palda (1993), (1999) for broader overviews and discussion.
The well-known calculus of voting model of turnout (Downs (1957), Riker
and Ordeshook (1968)) defines the benefits of voting as pB + d where p is the
probability of swinging the election, B is the gain from having one’s preferred
candidate win, and d is the benefit a citizen feels from doing his civic duty or
expressing his preference. A voter votes if these benefits exceed the direct cost
of voting, denoted c, which includes the time taken to get to the polls and so
on. To get a useful theory of turnout, it is necessary to understand how these
variables depend on election specific characteristics.
Since the benefits from doing one’s duty seem rather nebulous, it is tempting
to look at the pB term to understand turnout. The pivotal-voter model of
Ledyard (1984) and Palfrey and Rosenthal (1985) provides a natural way of
endogenizing the probability that a voter will swing the election. However,
the obvious problem with this approach would seem to be that p is sufficiently
small in any large election that changes in pB are likely to be minuscule across
elections. Thus, many have questioned the fruitfulness of a theory of turnout
based on minuscule changes in a minuscule number (see, for example, Green
and Shapiro (1994)). Formal support for this concern is provided by Palfrey
and Rosenthal’s well known result that in a sufficiently large electorate the only
citizens who vote in equilibrium are those for whom d is no smaller than c.3
3 Palfrey
and Rosenthal’s result is for symmetric equilibria in a model where voters are
4
Accordingly, significant variations in turnout in large elections must arise from
variations in the fraction of the population for whom d is no smaller than c.
More recently, researchers have turned to the d term. An interesting line of
work has assumed that this term can be influenced by leaders (see, for example,
Shachar and Nalebuff (1999)). The idea of the follow the leader model is that
in close elections or in elections where there is much at stake, community and
political leaders put in more effort exhorting their fellows to vote and this leads
to higher turnout. The effort decisions of political leaders are rational because
their efforts can sway large groups of voters. Exactly why such exhortions are
successful is not clear, which seems a difficulty with the approach.4
An alternative strategy is to think more deeply about where individuals’
notions of duty in the voting context may come from. Harsanyi (1980) argues
that voting may usefully be understood as individuals acting according to the
dictates of rule-utilitarianism - individuals follow the voting rule that would
maximize aggregate utility if everybody followed it. Harsanyi illustrates his
argument by considering an environment in which a fixed number of votes are
needed to pass a policy that would raise aggregate utility. Each citizen faces
the same cost of voting and chooses a probability of voting that, if adopted
by all, would maximize aggregate utility. The key insight is that the optimal
probability is between zero and one. Not everybody should stay home, because
that would mean the policy would not pass. But not everybody should vote
because that would result in a surfeit of votes, imposing unnecessary costs on
society. In this way, the logic of rule utilitarianism yields an elegant theory of
turnout. In terms of the calculus of voting model, Harsanyi effectively assumes
that d is large enough so that everyone does their duty but rejects the implicit
assumption that doing one’s duty always involves voting.
Harsanyi’s insight is developed much further by Feddersen and Sandroni
(2002). They consider the more relevant environment of a two candidate plurality rule election in which citizens have heterogeneous voting costs. Feddersen
and Sandroni first point out a problem with Harsanyi’s argument in this context. With two candidates to choose from, a rule utilitarian has to choose not
only whether to vote but also for whom to vote. All rule utilitarians would
vote for the candidate that maximizes aggregate utility and, accordingly, if only
rule utilitarians voted, the optimal voting rule would be such that turnout is
minimal. Since all voters would be voting for the same candidate, it is best for
society as a whole to minimize the number of individuals incurring voting costs.
To deal with this problem, Feddersen and Sandroni introduce disagreement
on which candidate maximizes aggregate utility. There are two groups of rule
utilitarians with opposing views. Individuals in each group follow a voting rule
that, if followed by all in their group, would maximize aggregate utility given
the behavior of individuals in the opposing group. Feddersen and Sandroni show
imperfectly informed about each others’ voting costs and preferences.
4 In one of the first papers stressing the importance of group leaders, Uhlaner (1989) assumed that leaders offered transfers to group members in exchange for their votes. However,
even when transfers are interpreted most broadly, this practice does not seem particularly
widespread.
5
that the two groups’ voting rules can be derived as the equilibrium of a game in
which group members chooses a rule for their group to maximize their version of
expected aggregate utility. While there are differences in the details, this game
has the same basic structure as the one studied in this paper.5
The key difference between Feddersen and Sandroni’s rule-utilitarian model
and the group rule-utilitarian model of this paper is that in the former individuals follow the voting rule that they believe maximizes aggregate utility, while
in the latter they follow the voting rule that maximizes the payoff of those on
their side of the issue. Using the terminology of social psychology, the distinction is one between “altruism” and “collectivism”.6 The merit of Feddersen
and Sandroni’s approach is that all behavior follows from the single postulate
that citizens are rule-utilitarians. This has significant theoretical appeal. However, the social psychology literature stresses the importance of group identity
for cooperation in social dilemma type of situations (see, for example, Dawes,
van de Kragt and Orbell (1988), Tajfel (1981) and Turner (1987)) and in contests (such as elections) being on the same side creates a natural group identity.
Moreover, it is not clear why citizens should have different beliefs, nor what
should determine the relative sizes of the two groups.7
Our group rule-utilitarian model is also related to the work of Morton (1987),
(1990). She studies a two candidate election and assumes that the population is
exogenously divided into groups with different policy preferences. Each group
collectively and simultaneously decides how many of its members should vote
in order to maximize the group’s aggregate benefit. The choice trades off the
policy benefit associated with changing the outcome of the election with the cost
to members of voting. In Coasian fashion, Morton is not specific on why the
groups behave in this way: “The model assumes that groups invest resources
(financial or otherwise) which provide group members with the individualized
incentives necessary to vote. These resources are then transformed into votes
by the groups.” (Morton (1987) page 120). This paper’s approach may be
considered as a special case of Morton’s in which there are only two groups supporters and opposers. While we prefer our group rule-utilitarian interpretation, there is nothing in the empirical work to distinguish it from a story where
supporters and opposers collectively determine which of their members should
5 There are three main differences in the details. First, in the model of this paper, the two
groups may differ in the intensity of their preference for their preferred candidates. Second,
in Feddersen and Sandroni’s model, the fraction of each group who behave “ethically” (i.e., as
rule utilitarians) is random. Non-ethical voters abstain. Third, in Feddersen and Sandroni’s
model, ethical voters will only follow the optimal rule if their payoff from ethical behavior (the
d term) exceeds their voting cost. As in Harsanyi (1980), the model of this paper implicitly
assumes that d is sufficiently large that individuals always do their part.
6 As defined by Batson (1994) in the Handbook of Social Psychology : “Collectivism involves
motivation to benefit a particular group as a whole. The ultimate goal is not one’s own welfare
or the welfare of the specific others who are benefited; the ultimate goal is the welfare of the
group” (p.303).
7 These problems make it difficult to use our data to directly test between the rule-utilitarian
and group rule-utilitarian models. However, it is possible to estimate a version of the rule
utilitarian model with our data and we explain how to do this in the Appendix.
6
vote.8
While there is a large empirical literature on turnout, there are very few papers that try to structurally estimate models of turnout. Hansen, Palfrey and
Rosenthal (1987) use data on school budget referenda to try to structurally estimate the pivotal-voter model. Given its complexity, they must make strong assumptions to undertake the estimation. In particular, they assume that the population is equally divided between supporters and opposers and that supporters
and opposers have identical benefits from their preferred outcomes. They then
estimate the parameters of the distribution of voting costs. Our simpler model
permits estimation of the distribution of supporters and opposers, the benefits
of supporters and opposers, and the distribution of voting costs.
Shachar and Nalebuff (1999) use state-by-state voting in U.S. presidential
elections to structurally estimate a model based on the “follow the leader” approach. Their model assumes that Democratic and Republican leaders in each
state expend effort to impact the outcome of the presidential election.9 Leaders’ ability to have an impact depends on how followers respond and on the
expected closeness of the race (at both state and national levels). The former
is a parameter of the model to be estimated and the latter depends on the distribution of Democrats and Republicans in the population, which is estimated
from past election outcomes. The authors conclude that voters do respond to
effort and that effort is higher in races that are predicted to be closer. Shachar
and Nalebuff’s model is an equilibrium model in that the leaders from the two
parties in each state choose their effort levels simultaneously. This gives it a
similar flavor to our model.
3
3.1
Preliminaries
Institutional background
Chapter 251 of the Texas Alcoholic Beverage Code states that “On proper
petition by the required number of voters of a county, or of a justice precinct or
incorporated city or town in the county, the Commissioners’ Court shall order
a local election in the political subdivision to determine whether or not the sale
of alcoholic beverages of one or more of the various types and alcoholic contents
shall be prohibited or legalized in the county, justice precinct, or incorporated
city or town”. Thus, citizens can propose changes in the liquor laws of their
communities and have their proposals directly voted on in referenda. Such
direct democracy has a long history in Texas liquor regulation, with local liquor
8 Morton (1987) endogenizes the policy choices office-seeking candidates would make given
this group voting behavior. Building on this work, Filer, Kenny and Morton (1993) present
a group voting model where candidates propose tax schemes that differ in the degree of
progressiveness. The paper empirically tests the model’s qualitiative predictions using countylevel turnouts in the 1948, 1960, 1968 and 1980 presidential elections.
9 One drawback with the study is that, in reality, voters are voting on many other issues
at the same time as they are casting their presidential ballot.
7
elections dating back to the mid 1800s.10
The process by which citizens may propose a change for their jurisdiction is
relatively straightforward. The first step involves applying to the Registrar of
Voters for a petition. This only requires the signatures of ten or more registered
voters in the jurisdiction. The hard work comes after receipt of the petition.
The applicants must get it signed by at least 35% of the registered voters in the
jurisdiction and must do this within thirty days.11 If this hurdle is successfully
completed, the Commissioners’ Court of the county to which the jurisdiction
belongs must order a referendum be held. This order must be issued at its
first regular session following the completion of the petition and the referendum
must be held between twenty and thirty days from the time of the order. All
registered voters can vote and if the proposed change receives at least as many
affirmative as negative votes, it is approved.
Citizens may propose changes for their entire county, their justice precinct, or
the city or town in which they reside. The state is divided into 254 counties and
each county is divided into justice precincts.12 Accordingly, a justice precinct
lies within the county to which it belongs. By contrast, a city may spillover
into two or more justice precincts. If only part of a city belongs to a particular
justice precinct that has approved a change, then that part must abide by the
new regulations. However, if the city then subsequently approved a different
set of regulations, then they would also be binding on the part contained in the
justice precinct in question. Effectively, current regulations are determined by
the referendum most recently approved. Over our data period, citizens almost
always choose to propose changes at the city or justice precinct level rather than
at the county level.
Importantly for the purposes of our study, liquor referenda are typically
held separate from other elections. Section 41.01 of the Texas Election Laws sets
aside four dates each year as uniform election dates.13 These are the dates when
presidential, gubernatorial, and congressional elections are held. In addition,
other issues are often decided on these days such as the election of aldermen, and
the approval of the sale of public land and bond issuances. Elections pertaining
to these other issues may occur, but rarely do, on dates other than uniform
election days. Liquor referenda, in contrast, do not typically occur on uniform
election dates. This reflects the tight restrictions placed by Chapter 251 on the
timing of elections.14
1 0 From 1919 to 1935, these elections were abolished as a result of prohibition. Since 1935,
the process of citizen-democracy has been governed by the procedures described in Chapter
251 of the Texas Alcoholic Beverage Code.
1 1 Prior to 1993, the number of signatures needed was 35% of the total number of votes cast
in the last preceding gubernatorial election.
1 2 The number of justice precincts in a county range from 1 to 8.
1 3 These are the third Saturday in January, the first Saturday in May, the second Saturday
in August and the first Tuesday after the first Monday in November.
1 4 Interestingly, the Texas state government voted in 2001 to require liquor law referendum
votes to occur on one of the four uniform election dates. This was to avoid the costs of holding
referenda separately.
8
3.2
Data
We assembled data on 363 local liquor elections in Texas between 1976 and 1996
where prior to the election the voting jurisdictions prohibited the retail sale of all
alcohol.15 Information on these elections was obtained from the annual reports
of the Texas Alcoholic Beverage Commission (TABC). These reports contain
the county, justice precinct, city or town voting on the referendum, the date
of the election, the proposed change, and the number of votes cast for and
against. As indicated in Table 1, the elections differed in the degree to which
restrictions were relaxed: 147 proposed permitting the selling of beer only or
beer and wine; 144 proposed permitting the sale of all alcoholic beverages for
off-premise consumption only (i.e., liquor stores); and 72 proposed not only that
all beverages be sold but they may be consumed off- and on-premise (i.e., bars
as well as liquor stores). Of these 363 elections, 2 were at the county level,
133 were at the justice precinct level and 228 were at the city or town level.
While not indicated in Table 1, at least one election occurred in 125 different
counties. While certain counties had multiple local liquor elections during this 20
year period, approximately two-thirds of the 363 elections involved jurisdictions
which account for a single election. For those jurisdictions that had multiple
elections, these elections often occur a number of years apart.
We supplemented our election data with information on county-, city- and
town-level populations, by age, obtained from the United States Census. Using
this information, we estimate the voting age population and the population over
the age of 50 at the time of an election. Table 1 indicates that the mean voting
age population in the 363 jurisdictions is 4,415 at the time of the elections.
We also attempted to find information that might tell us about the attitudes
of citizens towards the selling of alcohol. Using the county-level population and
county-level information on the number of adherents to Baptist denominations
from Churches & Church Membership in the United States, we constructed estimates of the fraction of the county population that is Baptist at the time
of an election. We use this fraction as a proxy for the fraction of Baptists in
each of the 363 jurisdictions; thereby, implicitly assuming that Baptists are uniformly distributed throughout each county. As indicated in Table 1, the average
fraction of the county population that is Baptist is 0.48.
Information about what type of alcohol could be sold and where it could be
consumed elsewhere in the county was obtained from the annual reports of the
TABC. The alcohol policy being voted on in a third of the elections is more
liberal than the alcohol policy in the rest of the county. Monthly information on
the number of alcohol related road accidents in each county was obtained from
the Texas Department of Public Safety. The average number of alcohol related
accidents per capita in a county for the twelve months prior to an election is
0.00204. Finally, whether the jurisdiction is located in a Metropolitan Statistical
Area (MSA) was determined using classifications obtained from the 1996 United
States Census. Table 1 shows that 44% of the elections involved jurisdictions
1 5 See the data appendix for a description of this data collection process. The appendix also
contains a detailed explanation of how certain variables are created.
9
located in an MSA.
In an attempt to get information about the costs of voting, daily weather
conditions at 44 weather stations in Texas was obtained from the United States
Carbon Dioxide Information Analysis Center (CDIAC). The weather conditions
on the day of each election are taken to be the same as those measured at a
weather station in close proximity to the voting jurisdiction. While many of
these elections occurred on rainy days, as expected, few occurred on days when
snow fell (see Table 1). Besides the weather, whether the election occurred
on a weekend and whether the election occurred in the summer are also likely
to affect the costs of voting. The majority of the 363 elections occurred on a
weekend while slightly more than a quarter occurred in June, July or August.
3.3
Some basic facts
Of our 363 referenda, 150 were approved and 213 were rejected by the voters.
The percent of the voting population that voted for the referendum averaged
17% across the 363 elections while the percent voting against averaged 19%.
The average turnout in these elections (calculated by dividing total votes by
voting age population) is 0.36 but there is substantial variation across elections.
Figure 1 presents the turnout information in a histogram where the vertical axis
measures the number of elections in each turnout category. While a number of
elections had turnout rates over 0.75, the majority had less than a third of the
voting age population vote. Interestingly, average turnout is significantly higher
in city-level elections (0.44 compared to 0.21). In addition, average turnout was
significantly higher when the referendum involved off-premise consumption of
all alcohol as opposed to just beer and wine or off- and on-premise consumption.
The elections tended to be close. When closeness is defined as the difference
between votes for and against divided by total votes, the average closeness is
0.25. The histogram of this measure is depicted in Figure 2 and demonstrates
that while the majority of the elections are relatively close, there are outliers.
Unlike turnout, the average value of this measure of closeness does not differ
significantly between city-level elections and justice precinct- or county-level
elections. It is also similar across the different types of regulatory changes.
It is natural to ask whether the data support the familiar idea that turnout
is higher in close elections. This all depends on how we measure closeness.
Proceeding as in Figure 2, there is a slight positive relationship (correlation coefficient of 0.12). This positive relationship is stronger if closeness is defined as
the difference between votes for and against divided by the voting age population
(correlation coefficient of 0.58). However, there is a negative relationship between turnout and closeness when closeness is defined as the difference between
votes for and against (correlation coefficient of -0.11).
10
4
The group rule-utilitarian model
Consider a community that is holding a referendum on relaxing liquor laws. For
analytical tractability, we adopt the fiction that the community has a continuum
of citizens. These citizens are divided into supporters and opposers. Each
supporter is willing to pay b for the relaxation, while each opposer is willing to
pay x to avoid it.
Each citizen knows whether he is a supporter or an opposer, but not the fraction of citizens in each category. However, all citizens know that the fraction of
supporters in the population, denoted µ, is the realization of a random variable
with range [0, 1] distributed according to the Beta Distribution.16 Thus, the
probability density function of the random variable is
h(µ; ν, ω) = µν−1 (1 − µ)ω−1 /B(ν, ω)
where ν and ω are parameters known by the citizens and B(ν, ω) is the Beta
function
Z 1
B(ν, ω) =
µν−1 (1 − µ)ω−1 dµ.
0
The expected fraction of supporters under this distributional assumption is
ν/(ν + ω) and the variance is νω/[(ν + ω)2 (ν + ω + 1)]. We will assume that
both ν and ω exceed 1 which implies that the density is hump shaped.
Citizens must decide whether or not to vote in the referendum. If they do,
supporters vote in favor and opposers vote against. Voting is costly, with each
citizen i facing a cost of voting ci where ci is the realization of a random variable
uniformly distributed on [0, c]. Citizens know c but do not observe the voting
costs of their fellows. We assume that individuals follow the voting rule that, if
followed by everyone else on their side, would maximize their side’s aggregate
utility. Each side’s optimal voting rule specifies a critical cost level below which
an individual should vote.
Letting the critical voting costs for the two groups be denoted by γ s and
γ o , if citizen i is a supporter he votes if ci ≤ γ s and if he is an opposer he
votes if ci ≤ γ o . The probability that a supporter votes is the probability that
γ s exceeds ci , which is γ s /c. Similarly, the probability that an opposer votes is
γ o /c. Thus, the referendum passes when µγ s /c > (1 − µ)γ o /c or, equivalently,
when µ > γ o /(γ s +γ o ). The probability that the referendum passes is therefore:
Z 1
h(µ; ν, ω)dµ.
π(γ s , γ o ; ν, ω) =
γo
γ s +γ o
Prior to the realization of his voting cost, a supporter’s expected payoff from
the critical level γ s given that the critical level of opponents is γ o is given by:
π(γ s , γ o ; ν, ω)b −
γ 2s
.
2c
1 6 Shachar and Nalebuff (1999) model uncertainty in the fraction of the population who are
Democrats in a similar way. However, they assume that the fraction of Democrats is the
realization of a random variable with a normal distribution. This has the obvious drawback
that it can take on values outside the interval [0, 1].
11
The first term represents the expected policy benefits stemming from the referendum passing and the second term represent expected voting costs, given that
a supporter will vote only if ci ≤ γ s .17 Similarly, an opposer’s expected payoff
from the critical level γ o given that supporters have critical level γ s is
−π(γ s , γ o ; ν, ω)x −
γ 2o
.
2c
Accordingly, we define a pair of critical levels (γ ∗s , γ ∗o ) to be an equilibrium if
γ ∗s ∈ arg max {π(γ s , γ ∗o ; ν, ω)b −
γ s ∈[0,c]
γ 2s
}
2c
and
γ ∗o ∈ arg max {−π(γ ∗s , γ o ; ν, ω)x −
γ o ∈[0,c]
γ 2o
}.
2c
(γ ∗s , γ ∗o )
is an interior equilibrium if both γ ∗s and γ ∗o are between 0
We say that
and c.
We are now able to establish the following result.18
Proposition 1 If (γ ∗s , γ ∗o ) is an interior equilibrium, then
√
√
1
c( x)ν ( b)ω+2
∗
√
)2 ,
γs = ( √
ν+ω
( x + b)
B(ν, ω)
and
√
√
1
c( x)ν+2 ( b)ω
√
)2 .
γ ∗o = ( √
( x + b)ν+ω B(ν, ω)
This proposition shows that, if there exists an interior equilibrium, it is unique
and, moreover, the equilibrium critical levels are related to the parameters in
a relatively simple way. As intuition would suggest, both critical levels are
increasing in the maximal voting cost c, although γ ∗s /c and γ ∗o /c are decreasing
in c. The critical level for each group is increasing in the gains or losses to that
group caused by the referendum passing.
While providing a nice characterization of the equilibrium, the proposition
leaves open the question of existence. There is no general guarantee that an
equilibrium will exist - the payoff functions of supporters and opposers are not
quasi-concave functions of their own critical cost levels. Indeed, it is not difficult to find parameter values for which no equilibrium exists.19 In such circumstances, one of the critical levels described in the proposition is not a best
1 7 Given
that a supporter will vote if and only if his voting cost is less than γ s , his expected
R
Rγ
γ2
voting costs are 0 s ci dcc i + γc 0 dcc i which simplifies down to 2cs .
s
1 8 The proofs of this and the next proposition are in the Appendix.
1 9 Feddersen and Sandroni (2001) deal with this existence problem in their model by assuming that the fraction of individuals in each group who behave “ethically” (i.e., according
to the the dictates of rule utilitarianism) is uncertain. Under the assumption that the two
groups care equally intensely about the election, Feddersen and Sandroni show that equilibrium exists and is unique if the fraction of “ethicals” in each group is uncertain, independent,
and uniformly distributed.
12
response for the group in question. This is typically because it would be better
for that group not to vote at all than to vote for cost levels below the critical
level identified in the proposition. This arises, for example, when one group
(say, supporters) is expected to be much smaller than the other (i.e., ν/(ν + ω)
is small). While the cost level in the proposition is always positive and implies
a positive level of turnout, if supporters are very unlikely to win they may be
better off just giving up and staying home. But then if supporters are staying
home, the optimal critical cost for opposers becomes very small, which then
provides supporters an incentive to vote.
When can we be sure that a pair of critical cost levels satisfying the conditions of Proposition 1 is actually an equilibrium? Our next proposition provides
some useful sufficient conditions.
Proposition 2 Suppose that (γ ∗s , γ ∗o ) ∈ (0, c]2 satisfies (i) the conditions of
Proposition 1, (ii) the “second order” conditions
(ν + 2)γ ∗s > (ω − 2)γ ∗o & (ω + 2)γ ∗o > (ν − 2)γ ∗s ,
and (iii) the “better than staying home” conditions
π(γ ∗s , γ ∗o ; ν, ω)b ≥ (γ ∗s )2 /2c & (1 − π(γ ∗s , γ ∗o ; ν, ω))x ≥ (γ ∗o )2 /2c.
Then (γ ∗s , γ ∗o ) is an equilibrium.
The second order conditions in (ii), together with the conditions of Proposition
1, imply that the payoff functions of supporters and opposers are locally strictly
concave at (γ ∗s , γ ∗o ). The better than staying home conditions in (iii) ensure that
at (γ ∗s , γ ∗o ) the payoffs of supporters and opposers are at least as high as if they
simply choose not to vote. The proof of the proposition amounts to showing
that the payoff functions can have at most one interior local maximum in which
case these three conditions are sufficient to imply that γ ∗s is a best response to
γ ∗o and vice versa.
In our empirical work, we estimate the determinants of the exogenous variables (b, x, c, ν, ω) assuming that supporters and opposers use the critical levels
described in Proposition 1. Of course, this is only legitimate if these are indeed
equilibrium critical levels. We can check this using Proposition 2. Our estimates imply values of the exogenous variables (b, x, c, ν, ω) for each jurisdiction
which, in turn, imply values of the critical costs (γ ∗s , γ ∗o ) via the equations of
Proposition 1. If these implied values satisfy the second order conditions and
the better than staying home conditions of Proposition 2, then we know that
(γ ∗s , γ ∗o ) really are equilibrium critical costs given (b, x, c, ν, ω).
5
Estimation
We assume that for each jurisdiction j, ν j = 1 + exp(β v · zvj ) and ω j = 1 +
exp(β ω ) where β ν is a vector of parameters to be estimated, zvj is a vector
of jurisdiction specific characteristics that may influence the mix of supporters
13
and opposers, and β ω is a parameter to be estimated. We further assume
that xj = exp(β x · zxj + εj ) and bj = exp(β b · zbj + εj ) where β x and β b are
vectors of parameters to be estimated, zxj and zbj are vectors of jurisdiction
and referendum specific characteristics that may affect supporters’ benefits and
opposers’ costs and εj is the realization of some random variable distributed
according to the standard normal distribution. The assumption that εj is a
common shock to both supporters’ benefits and opposers’ costs allows us to
derive the likelihood function. Finally, we assume that cj = exp(β c · zcj ) where
β c is a vector of parameters to be estimated and zcj is a vector of jurisdiction
and referendum specific variables that may impact voting costs. The functional
forms are selected to ensure that the Beta distribution parameters ν j and ω j
are greater than one and that xj , bj , and cj are non-negative.
Our task is to estimate the parameters Ω = { β ν , β ω , β x , β b , β c }. To
construct the likelihood function, fix Ω and consider a particular jurisdiction j.
Suppose that the fraction of the population voting in favor of the referendum
is vsj and against is voj . Then, according to the model, vsj = µj γ ∗sj /cj and
voj = (1 − µj )γ ∗oj /cj where µj is the fraction of the voting population who are
supporters and γ ∗sj and γ ∗oj are the equilibrium critical cost levels for supporters
and opposers.
Using the formulas presented in Proposition 1, it follows that
p
√
µj
cj ( xj )ν j ( bj )ωj +2
1
p
)2
vsj =
( √
ν
+ω
j
j
cj ( xj + bj )
B(ν j , ω j )
and that
voj
p
√
cj ( xj )ν j +2 ( bj )ωj
(1 − µj )
1
p
)2
=
( √
cj
( xj + bj )ν j +ωj B(ν j , ω j )
Substituting in our functional forms and rearranging, we can write these as:
√
vsj = µj exp εj Kj
and
voj
p
x
bj
√
= (1 − µj ) exp εj Kj ( q ),
bbj
bj = exp(β x · zxj ) and
where bbj = exp(β b · zbj ), x
p ν q ω +2
bj ) j ( bbj ) j
( x
1
q
)2 .
Kj = ( p
( x
bj + bbj )ν j +ωj B(ν j , ω j )cj
We can now solve these two equations for the realizations of µj and εj implied
by any given choice of parameters Ω. In this way, we obtain:
p
vsj x
bj
q
µj =
p
voj bbj + vsj x
bj
14
and
εj = 2 ln(voj
q
p
p
bbj + vsj x
bj ) − 2 ln(Kj x
bj ).
These equations define µj and εj as functions of the turnouts (vsj , voj ). Using the distributions of µj and εj , we can now compute the probability of observing any pair of turnouts (see the Appendix for the derivation). Letting
Zj = (zxj , zbj , zvj , zcj ), the probability density function for (vsj , voj ) is
g(vsj , voj
where
ςj
q
p ν
bj ) j (vsj )ν j −1 ( bbj )ωj (voj )ωj −1
2( x
ς 2j
1
q
√
·
| Ω, Zj ) =
exp(−
),
p
2
2π
(voj bbj + vsj x
bj ))ν j +ωj B(ν j , ω j )
q
p
= (ν j + ω j ) ln( x
bj + bbj ) + ln B(ν j , ω j ) + ln cj
q
q
p
p
+2 ln(voj bbj + vsj x
bj ) − (ν j + 2) ln x
bj − (ω j + 2) ln bbj .
Accordingly, our likelihood function is
L(Ω) =
J
Y
j=1
g(vsj , voj | Ω, Zj ).
(1)
Any given estimate of the parameters Ω = { β ν , β ω , β x , β b , β c } implies
values of the exogenous variables for each jurisdiction j.20 These, in turn,
imply values of the critical costs via the equations of Proposition 1. Unconstrained maximization of the likelihood function generates parameter estimates
which, for some jurisdictions, imply values of the critical costs that exceed the
maximum possible cost. Since this is clearly inconsistent with the model, we
must maximize the likelihood function subject to the feasibility constraints that
γ ∗sj ≤ cj and γ ∗oj ≤ cj for each jurisdiction j.21
To see how to impose these constraints, observe that for each jurisdiction j
q
p
bbj + vsj x
∗
bj
v
γ sj
oj
vsj
p
=
=
,
cj
µj
x
bj
2 0 Thus,
ν j = 1 + exp(β v · zvj ), ω j = 1 + exp(β ω ), and cj = exp(β c · zcj ). Moreover, xj =
q
p
p
bj )−2 ln(Kj x
bj ).
exp(β x ·zxj +εj ) and bj = exp(β b ·zbj +εj ) where εj = 2 ln(voj bbj +vsj x
2 1 By imposing these feasibility constraints, we are requiring that the choice of parameters
must satisfy the conditions of Proposition 1 when either γ ∗sj and γ ∗oj equals cj . This restricts
the choice of parameters in a marginally tighter way than is implied by the model. This is
because the conditions of Proposition 1 need not be satisfied if either group’s critical cost level
is at the boundary. In the boundary case, the first order conditions are in the form of weak
inequalities rather than equalities. Since this dampens the ability of the model to fit the data,
it will in no way compromise our conclusions about the relative performance of the model.
15
and that
voj
γ ∗oj
voj
=
=
cj
1 − µj
q
p
bbj + vsj x
bj
q
.
bbj
Using these, the feasibility constraint for jurisdiction j can be written as
(
vsj 2 bbj
1 − vsj 2
) ≤
≤(
) .
1 − voj
x
bj
voj
Substituting in for bbj and x
bj , yields
ln(
vsj 2
1 − vsj 2
) ≤ β b · zbj − β x · zxj ≤ ln(
) .
1 − voj
voj
(2)
We can now solve for the parameters that maximize the likelihood function
subject to these constraints.22 As noted in the previous section, we may then
use Proposition 2 to check whether the estimated critical cost levels are actually
an equilibrium given the values of the exogenous variables. Happily, this is the
case for every jurisdiction.
6
Results
The empirical results of the basic model are shown in Tables 2 and 3. Table
2 presents the parameter estimates that maximize the likelihood function in
Equation (1) subject to the constraints specified in Equation (2). Using these
parameter estimates, Table 3 presents some aggregate information about the
implied values of the model’s exogenous variables for the 363 jurisdictions.
We expect the fraction of supporters in a jurisdiction to be a function of
the fraction of baptists, the fraction of citizens over the age of 50, the number
of alcohol related accidents in the prior year, and the jurisdiction’s perception
of the effects of a less restrictive alcohol policy. This perception is likely to
depend on whether the jurisdiction is a city (compared to a justice precinct or
county) and whether the jurisdiction is located in an MSA. Therefore, we allow
the model’s Beta distribution to vary across jurisdictions by specifying ν as a
function of these five variables.
The coefficient estimates in Table 2 imply that the average expected percentage of supporters across the 363 jurisdictions is 5.07/(5.07+4.14)=55% (see
Table 3). The estimates also indicate that increasing by ten percent the fraction
of baptists or the fraction of voting age population over the age of 50, decreases
supporters by approximately one percent. This suggests that both baptists and
people over 50 years old are ten percent more likely to oppose the referendum
2 2 The Appendix contains a detailed description of how we solved the constrained maximization problem.
16
than non-baptists and people under the age of 50, respectively.23 The number of prior year’s alcohol related accidents in the county increases the fraction
of supporters. This is consonant with Baughman, Conlin, Dickert-Conlin and
Pepper (2000) who find that the number of alcohol related accidents in Texas
counties may actually decline with a less restrictive alcohol policy.24 In addition, the fraction of supporters is slightly less in cities and significantly less in
jurisdictions located in an MSA. The large negative coefficient associated with
the MSA variable implies that the fraction of supporters is eight percent less,
on average, if the jurisdiction is located in an MSA. The reason for this might
be that a jurisdiction that allows the sale of alcohol attracts more outsiders if
it is in an urban compared to a rural area. While some residents may perceive
this as a benefit, others may be concerned about the type of people the alcohol
would attract. Alternatively, it may be that black market liquor is more readily
available in urban areas.
We allow supporters’ benefit and opposers’ cost of a passed referendum to
depend on the type of referendum, whether the jurisdiction voting is a city and
whether passing the referendum would result in the jurisdiction having a more
liberal alcohol policy than any other jurisdiction in the county. The coefficient
estimates indicate that all of these factors have large and statistically significant effects. As for the type of referendum, Table 2 shows that the supporters’
benefit and opposers’ costs are greater when the vote pertains to off-premise
consumption of all alcohol than when it involves beer and wine. The average
marginal effects are to increase the supporter’s benefit by 0.17 and the opposers’
costs by 0.10. These are relatively large given the average benefits and costs
to supporters and opposers (see Table 3). While the positive coefficients associated with off-premise consumption were expected, the negative coefficients
associated with off- and on-premise consumption of all alcohol were not.
The positive coefficients associated with a referendum being voted on by a
city suggests that the benefits and costs to voters of a less restrictive alcohol
policy are greater in city-level elections. A possible explanation is that residents
in more densely populated areas (such as cities compared to justice precincts)
are more likely to feel the impact of an alcohol policy liberalization. Referenda
involving a more liberal alcohol policy than exists in the rest of the county also
increases supporters’ benefits and opposers’ costs. We expect these positive effects since if surrounding jurisdictions are tightly regulated, this should increase
the impact of a relaxation. As for the average marginal effects of a city referendum and a referendum involving the most liberal policy in the county, they
are 0.49 and 0.24 for the supporters’ benefit and 0.66 and 0.40 for the opposers’
costs, respectively.
2 3 The fact that these variables are measured at the county-level and religious affiliations
are available at the county level only in 1970, 1980 and 1990, makes these noisy measures of
the fraction of baptists and the fraction of people over the age of 50 in the jurisdiction at the
time of the election. This may explain why their coefficients are not statistically significant.
2 4 While these law changes decrease the implicit price of alcohol for the jurisdiction, they
also reduce the travel distance required to obtain the alcohol. Baughman, Conlin, DickertConlin and Pepper (2000) find that for certain alcohol policy liberalizations, this second effect
dominates in regards to alcohol related accidents.
17
The coefficients in Table 2 indicate that while the cost of voting does depend
on whether the election is held on the weekend, the weather conditions on the
day of the election and summer-time elections do not significantly effect the cost
of voting. The positive coefficient of 0.282 associated with the weekend indicator
variable implies an average marginal effect of 0.65 on the upper support of voting
costs. We felt that voting on a weekend might be more costly because individuals
would have to give up their leisure time in order to vote. The marginal effect is
significant given that the average upper support of voting costs across the 363
jurisdictions is 2.45 (see Table 3).
The average values of bj and xj presented in Table 3 indicate that opposers
feel much more intensely about the issue than do supporters. This greater
intensity translates into opposers having significantly higher critical cost levels
on average than supporters. The average critical cost level is 0.76 for supporters
and 0.96 for opposers. These yield average turnout rates of 31% for supporters
and 42% for opposers.
It is important to note that the average values of bj , xj and cj presented in
Table 3 are in relative terms and the numerical values have no significance.25
However, a feel for the numbers in dollar terms, can be obtained by using
intuition to assign a value to the cost of voting. For example, suppose that
we guess that the average voting cost across all districts is $15. Then, this
implies that c/2 = 2.45/2 = $15. This ties down the units in dollar terms since
2.45 = $30 or 1 = $30/(2.45) = $12.245. It follows that the average value of x
is $12.245(0.92) = $11.26 and the average value of b is $12.245(0.59) = $7.22.
These numbers suggest that the proposed regulatory changes are of rather minor
importance to citizens’ welfare.
The predictions of the model’s exogenous variables and reasonable cost levels
imply a value of µj for each district. This can be combined with bj and xj to
provide a measure of the average net benefit of the proposed change µj bj − (1 −
µj )xj . The change passes a standard cost-benefit test if and only if this average
net benefit is positive. Of the 363 jurisdictions, 110 had a positive net benefit.
While all of these 110 referenda did pass, 49 of the referenda with a negative
net benefit also passed. This shows that there is no reason to believe that group
rule-utilitarian voting implies surplus maximizing outcomes.
A proposed change with a positive net benefit does not imply that holding
a referendum is desirable because of the transactions costs associated with voting. Holding the referendum passes a cost-benefit test if and only if µj (bj −
(γ ∗sj )2 /2cj ) − (1 − µj )(xj + (γ ∗oj )2 /2cj ) is positive. Only 39 referenda had a
positive net benefit when voting costs are included.26 This suggests that the
2 5 Note from Table 2 that there is no constant term in the cost of voting function. By not
including a constant term we are setting the maximal voting cost equal to one for elections
held on a non-summer weekday where there is no rain nor snow and the temperature is zero
degrees Fahrenheit. The normalization is required because we cannot infer benefits and costs
from the number of people who vote for and against the referendum. Instead, we can only
infer relative benefits and relative costs.
2 6 The Texas state government’s recent move to require that liquor referenda be held on
uniform election dates should help in this respect by spreading the transactions costs over a
number of ballot issues. However, since it will also impact the likely turnout pattern, it may
18
case for this form of direct democracy is weak when evaluated on conventional
cost-benefit grounds.
The following section presents two alternative models by which to test the
appropriateness of the current model. Because the comparison of the three models will be primarily based on how well the models fit the data, it is interesting
to note that the model just estimated explains 54 percent of the variation in
turnout for the referenda and 49 percent of the variation in turnout against the
referenda.
7
Alternative models
The previous section showed how to estimate the parameters of our turnout
model assuming that it was the correct model. While the fact that the parameter
estimates seem reasonable is comforting, the results give us no reason to believe
that our basic model is a good approximation of voting behavior. To provide
evidence on this, we compare it with two simple alternatives. In the framework
of the calculus of voting model, these alternatives can be thought of as providing
different accounts of what determines the d term - the non-instrumental benefit
from voting.
7.1
The intensity hypothesis
The intensity hypothesis asserts that people get a higher payoff from voting the
more intensely they feel about an issue. This is consistent with an expressive
view of voting (see, for example, Brennan and Lomasky (1993)). Voting is like
cheering at a football game and you are more likely to cheer the more you care
about the outcome. Formally, we assume that supporters vote if their voting
cost is less than
γ s = αb
and opposers vote if their voting cost is less than
γ o = αx,
where α > 0. Here, the parameter α measures the strength of citizens’ desire to
express themselves through voting which may depend upon community characteristics. The key restriction is that both supporters and opponents share the
same α. Under this specification, the probability that a supporter votes is the
probability that γ s exceeds his voting cost, which is γ s /c = αb/c. Similarly, the
probability that an opposer votes is γ o /c = αx/c.
To estimate the model, we assume as in the group rule-utilitarian model
that for each jurisdiction j, ν j = 1 + exp(β v · zvj ), ω j = 1 + exp(β ω ), xj =
exp(β x ·zxj +εj ), bj = exp(β b ·zbj +εj ) and cj = exp(β c ·zcj ). We further assume
that αj = exp(β α · zαj ) where β α is a vector of parameters to be estimated and
zαj is a vector of jurisdiction specific characteristics that may affect citizens’
also increase the set of referenda with negative net benefits that pass.
19
desires to express themselves. The parameters to be estimated are Ω = {
β v , β ω , β b , β x , β α , β c }.
To construct the likelihood function, fix Ω and consider a particular jurisdiction j. Then, according to the intensity hypothesis,
vsj = µj exp εj
and
αjbbj
,
cj
voj = (1 − µj ) exp εj
αj x
bj
,
cj
bj = exp(β x ·zxj ). Letting Zj = (zvj , zbj , zxj , zαj , zcj ),
where bbj = exp(β b ·zbj ) and x
the probability density function for (vsj , voj ) is then27
g(vsj , voj ; Ω, Zj ) =
where
ς 2j
1
(b
xj )ν j (vsj )ν j −1 (bbj )ωj (voj )ωj −1
· √ exp(− ),
2
2π
(bbj voj + x
bj vsj )ν j +ωj B(ν j , ω j )
bj vsj ) − ln αj − ln x
bj − ln bbj .
ς j = ln cj + ln(bbj voj + x
As in the group rule-utilitarian model, we must ensure that the parameter
estimates are such that γ sj and γ oj are less than the upper support of the
distribution of voting costs. Therefore, we maximize the likelihood function
subject to the constraints that γ sj ≤ cj and γ oj ≤ cj for each jurisdiction j.
These constraints impose the following restrictions on the parameter estimates:
for all jurisdictions j,
ln(
vsj
1 − vsj
) ≤ β b · zbj − β x · zxj ≤ ln(
).
1 − voj
voj
Table 6 shows the parameter values that maximize the likelihood function subject to the above constraints and Table 7 contains average implied values of the
exogenous variables (ν j , ω j , αj , bj , xj , cj ) based on the parameter estimates. We
use the same variables to explain variation in ν, b, x, and c as in the group ruleutilitarian model. The coefficients in Table 6 suggest that the effects of these
variables are similar to those in the group rule-utilitarian model. As indicated
by the implied values in Table 7, the average expected percentage of supporters
predicted by the intensity model is almost identical to that in the group ruleutilitarian model (54% compared to 55% ). While it is impossible to directly
compare the implied values of b, x and c across models, note that while in both
models the implied value of the opposers’ cost is greater than the supporters’
benefit, on average, this difference is much larger in the group rule-utilitarian
model.
The strength of citizens’ desire to express themselves through voting is likely
to vary across jurisdictions depending on the jurisdiction’s religious composition,
2 7 Again,
the Appendix provides the derivation.
20
age distribution, and size. The coefficients in Table 6 associated with the fraction
of county population that is baptist, the fraction of county voting age population
over 50, and voting age population suggest that individuals who are baptist, over
the age of 50, and reside in smaller jurisdictions have a stronger desire to express
themselves through voting.
7.2
The popularity hypothesis
The popularity hypothesis asserts that people are more willing to vote if they expect that many of their fellow citizens share their position on the issue. The idea
is that the returns from voting are in the form of social approval (or avoidance
of disapproval) from those who share one’s position (as discussed, for example,
by Coleman (1990)). Accordingly, a supporter failing to vote for the referendum
when most other citizens are supporters, will experience more disapproval than
if most citizens are opponents.
ν
Recalling that the mean value of µ is ν+ω
, we can capture this idea by
assuming that supporters vote if their voting cost is below
γs = α
ν
,
ν +ω
while opposers vote if their voting cost is below
γo = α
ω
,
ν+ω
where α > 0. This says that individuals from each group are more likely to
vote the larger is the expected proportion of their group in the population. The
parameter α measures the value of obtaining approval in the election. This will
depend on community characteristics (such as size) and also on the salience of
the election to citizens. Again, the key restriction is that it is the same for
both supporters and opponents. Under this specification, the probability that
a supporter votes is the probability that γ s exceeds his voting cost, which is
γ s /c = αν/c(ν + ω). Similarly, the probability that an opposer votes is γ o /c =
αω/c(ν + ω).
To estimate the model, we assume as in the basic model that for each jurisdiction j, ν j = 1+exp(β v ·zvj ), ω j = 1+exp(β ω ) and cj = exp(β c ·zcj ). We also
assume that αj = exp(β α · zαj + εj ) where β α is a vector of parameters to be
estimated, zαj is a vector of jurisdiction and referendum specific characteristics
that may affect the value of obtaining approval and εj is the realization of some
random variable distributed according to the standard normal distribution. The
vector of parameters to be estimated is therefore Ω = { β ν , β ω , β α , β c }.
To construct the likelihood function, fix Ω and consider a particular jurisdiction j. Then, according to the popularity hypothesis,
vsj = µj exp εj
α
bj
νj
·
,
cj ν j + ω j
21
and
voj = (1 − µj ) exp εj
ωj
α
bj
·
,
cj ν j + ω j
where α
b j = exp(β α · zαj ). Letting Zj = (zαj , zvj , , zcj ), the probability density
function for (vsj , voj ) is then
g(vsj , voj ; Ω, Zj ) =
ς2
(ω j )ν j (vsj )ν j −1 (ν j )ωj (voj )ωj −1
1
√ exp(− j ),
·
ν
+ω
(ν j voj + ω j vsj ) j j B(ν j , ω j )
2
2π
where
b j + ln cj − ln vj − ln ω j .
ς j = ln(vj voj + ω j vsj ) + ln(vj + ω j ) − ln α
The constraints that γ sj ≤ cj and γ oj ≤ cj for each jurisdiction j impose the
following restrictions on the parameter estimates: for all j
vsj
1 + exp(β v · zvj )
1 − vsj
≤
.
≤
1 − voj
1 + exp(β ω )
voj
Tables 8 and 9 show the optimal parameter values and the implied average
values of the exogenous variables (ν j , ω j , αj , cj ).28 We use the same variables to
explain variation in ν and c as in the group rule-utilitarian model. As indicated
by the implied values in Table 9, the average expected percentage of supporters
predicted by the popularity model is 49% which is less than the 55% predicted by
the group rule-utilitarian model. Besides that associated with the city indicator
variable, the coefficients in Table 9 suggest that the effects of the variables on the
fraction of supporters are similar to those in the group rule-utilitarian model.
The positive and statistically significant coefficient of 0.085 in the popularity
model indicates that the fraction of supporters increases by two percentage
points if the election involves a city. As for the cost of voting, the only coefficient
estimate in Table 8 that is appreciably different than the analogous estimate in
Table 2 is the one associated with the weekend indicator variable. While both
models predict that having an election on a weekend increases the cost of voting,
the predicted effect is larger for the group rule-utilitarian model. In addition,
the popularity model predicts a much lower average cost of voting than the
group rule-utilitarian model.
The social approval supporters and opposers obtain from voting is likely
to depend not only on the referendum but also the characteristics of the jurisdiction. We expect the social approval from voting to be greater the more
important the referendum and the more individuals interact with others who
have similar views regarding the liquor referendum. The coefficients in Table 8
indicate that social approval increases when the referendum involves off-premise
consumption of alcohol, when the referendum involves a city, and when the surrounding jurisdictions have more restrictive alcohol policies. While these effects
are as expected, the negative coefficient associated with the off- and on-premise
2 8 The
parameter estimates in Table 6 are such that none of the constraints bind.
22
consumption is surprising. Interestly, the direction and magnitude of these effects on social approval are similar to those on supporters’ benefit and opposers’
costs in the group rule-utilitarian model. As for the size of the jurisdiction, the
negative and statistically significant coefficient of -0.024 associated with voting
age population suggests that increasing a jurisdiction’s voting age population
by 5,000 increases the social approval from voting by, on average, 11.5 percent.
7.3
Comparing the models
To test the validity of the group rule-utilitarian model relative to the intensity
and popularity models, we use the directional test for non-nested models proposed by Vuong (1989). Vuong proposes a likelihood-ratio based statistic to test
the null hypothesis that two competing models are equally close to the true data
generating process against the alternative hypothesis that one model is closer.
Vuong proves that the difference between the maximum log-likelihood values of
Model A and Model B divided by the product of the standard deviation of the
difference in the log likelihood value for each observation and the square root of
the number of observations has a standard normal distribution if the two models
are equivalent. Vuong also demonstrates that the null hypothesis that Models
A and B are equivalent can be rejected when the alternative hypothesis is that
Model A (B) is better than Model B (A) if the above test statistic is greater
(less) than critical value c (-c) obtained from the standard normal distribution
for some significance level.
The maximum log-likelihood values for the group rule-utilitarian, intensity,
and popularity models are 749.20, 704.05, and 692.26 respectively. Table 8
presents the value of Vuong’s test statistic for the possible null hypotheses.
These values indicate the null hypothesis that the group rule-utilitarian model
is equivalent to the popularity model and that the group rule-utilitarian model is
equivalent to the intensity model can be rejected at the five percent significance
level when the alternative hypothesis is that the group rule-utilitarian model is
better. However, these null hypotheses cannot be rejected when the alternative
hypotheses are that the popularity and intensity models are better, respectively.
Table 8 also indicates that the null hypothesis that the popularity and intensity
models are equivalent can be rejected at the five percent significance level when
the alternative hypothesis is that the intensity model is better.
8
Conclusion
This paper has made use of a unique data set to structurally estimate a group
rule-utilitarian model of voter turnout and to statistically compare it with two
simple alternatives. The results are encouraging: the structural estimation
yields reasonable coefficient estimates and the model performs better than the
alternatives. This suggests that the approach to thinking about turnout that
underlies the model warrants serious consideration.
23
There are many different directions for future research on this general approach (see also Feddersen and Sandroni (2002)). From an empirical perspective, it would be worth comparing the performance of the group rule-utilitarian
model with that of the pivotal-voter model. While there are good reasons to
be sceptical about its abilities to explain turnout, the pivotal voter model represents in many respects the simplest way of thinking about voting behavior.
Thus, it should only be rejected if it can be shown to be outperformed by some
coherent alternative. This has yet to be demonstrated. The data set used in
this paper is appropriate for studying the pivotal-voter model and the model we
have developed is a coherent alternative. It remains to structurally estimate the
pivotal-voter model and compare its performance. Given its greater complexity,
this will be a challenging task.
With our data, it is not possible to directly compare the performance of
our group rule-utilitarian model and the rule-utilitarian model of Feddersen
and Sandroni (2002). Future research could try and distinguish between these
models in experiments. It would seem possible to set up an experiment where
participants are assigned to two groups who must participate in an election.
The two groups could be assigned different private benefits from the election
outcome as well as different beliefs concerning aggregate benefits. Individuals
could then be assigned different voting costs and one could see which model
best describes their voting behavior.
From a theoretical perspective, it would be interesting to think about the
implications of heterogeneity in supporters’ and opposers’ preferences. It seems
likely that, within groups, those voters who care less intensely about an issue will
have lower critical cost levels. This may reflect considerations of equity in the
allocation of the costs of voting. Another interesting topic is how to think about
elections with three or more candidates. While such elections naturally divide
the population into groups of supporters, it is no longer obvious how supporters
of an underdog candidate should vote. This is particularly the case when there
are differences among group members in their second choice candidate. Finally,
more thought should be given to the justification of the behavior postulated
here. Why should we expect citizens to behave as group rule-utilitarians in
elections?
24
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of Political Science, 37, 246-78.
[2] Aldrich, John, [1997], “When is it Rational to Vote,” in Mueller, Dennis (ed), Perspectives on Public Choice, Cambridge: Cambridge University
Press.
[3] Batson, C. Daniel, [1994], “Altruism and Prosocial Behavior,” in Lindzey,
Gardner and Aronson, Ed (ed), Handbook of Social Psychology, Vol. 2 (4th
Edition), New York: McGraw-Hill.
[4] Baughman, Reagan; Conlin, Michael; Dickert-Conlin, Stacy and John Pepper, [2000], “Slippery when Wet: The Effects of Local Alcohol Access Laws
on Highway Safety,” Journal of Health Economics, forthcoming.
[5] Brennan, Geoffrey and Loren Lomasky, [1993], Democracy and Decision:
The Pure Theory of Electoral Preference, Cambridge: Cambridge University Press.
[6] Coleman, James S., [1990], Foundations of Social Theory, Cambridge MA:
Harvard University Press.
[7] Dawes, Robyn; van de Kragt, Alphons and Orbell, John, [1988], ”Not Me or
Thee but We: The Importance of Group Identity in Eliciting Cooperation
in Dilemma Situations: Experimental Manipulations,” Acta Psychologica,
68, 83-97.
[8] Downs, Anthony, [1957], An Economic Theory of Democracy, New York:
Harper and Row.
[9] Feddersen, Timothy J. and Alvaro Sandroni, [2002], “A Theory of Participation in Elections,” mimeo, University of Rochester.
[10] Filer, John; Kenny, Lawrence and Rebecca Morton, [1993], “Redistribution,
Income and Voting,” American Journal of Political Science, 37, 63-87.
[11] Fiorina, Morris, [1997], “Voting Behavior,” in Mueller, Dennis (ed), Perspectives on Public Choice, Cambridge: Cambridge University Press.
[12] Green, Donald and Ian Shapiro, [1994], Pathologies of Rational Choice
Theory: A Critique of Applications in Political Science, New Haven: Yale
University Press.
[13] Grossman, Gene and Elhanan Helpman, [2001], Special Interest Politics,
Cambridge MA: MIT Press.
[14] Hansen, Stephen; Palfrey, Thomas and Howard Rosenthal, [1987], “The
Downsian Model of Electoral Participation: Formal Theory and Empirical
Analysis of the Constituency Size Effect,” Public Choice, 52, 15-33.
25
[15] Harsanyi, John C., [1980], “Rule Utilitarianism, Rights, Obligations and
the Theory of Rational Behavior,” Theory and Decision, 12, 115-33.
[16] Ledyard, John O., [1984], “The Pure Theory of Large Two-Candidate Elections,” Public Choice, 44, 7-41.
[17] Matsusaka, John and Filip Palda, [1993], “The Downsian Voter Meets the
Ecological Fallacy,” Public Choice, 77, 855-78.
[18] Matsusaka, John and Filip Palda, [1999], “Voter Turnout: How Much Can
we Explain?” Public Choice, 77, 855-78.
[19] Morton, Rebecca B., [1987], “A Group Majority Voting Model of Public
Good Provision,” Social Choice and Welfare, 4, 117-31.
[20] Morton, Rebecca B., [1991], “Groups in Rational Turnout Models,” American Journal of Political Science, 35, 758-76.
[21] Palfrey, Thomas and Howard Rosenthal, [1985], “Voter Participation and
Strategic Uncertainty,” American Political Science Review, 79, 62-78.
[22] Riker, William H. and Peter Ordeshook, [1968], “A Theory of the Calculus
of Voting,” American Political Science Review, 62, 25-42.
[23] Shachar, Ron and Barry Nalebuff, [1999], “Follow the Leader: Theory and
Evidence on Political Participation,” American Economic Review, 89, 52547.
[24] Tajfel, H, [1981], Human Groups and Social Categories: Studies in Social
Psychology, Cambridge: Cambridge University Press.
[25] Turner, John, [1987], Rediscovering the Social Group: A Self-Categorization
Theory, London: Basil Blackwell.
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Groups,” American Journal of Political Science, 33, 390-422.
[27] Vuong, Quang H., [1989], “Likelihood Ratio Tests for Model Selection and
Non-nested Hypotheses,” Econometrica, 57, 307-333.
26
9
9.1
Appendix
Information on the data
A total of 526 local liquor elections are identified in the annual reports of the
Texas Alcoholic Beverage Commission between 1976 and 1996. We use 363 of
these in our estimation. Of the 163 elections we do not use, 64 were missing
critical information29 and 43 involved elections where other items seem likely
to have been voted on at the same time.30 To keep the basic issue constant
across elections, we focus on proposals to move from a completely “dry” status
where the selling of any alcohol is prohibited at the retail level.31 Therefore,
we eliminate the 53 elections where the jurisdiction was not “dry” prior to the
election. Finally, in order to structurally estimate our model, we drop three
elections where zero votes were cast against the referendum.32
The United States Census Bureau provides annual county-level populations,
by age. This information allows us to determine the population and the fraction
of the population over 50 at the time of the election when the jurisdiction
voting is either an entire county or a justice precinct. The population of a
justice precinct at the time of an election is estimated by dividing the county
population by the number of justice precincts in the county. We expect this to
be a relatively good approximation based on information provided by the Texas
Legislative Council indicating that justice precincts are selected so that each
in a particular county has roughly the same number of residents. The fraction
of the justice precinct over the age of 50 is assumed to be the same as in the
county.
In addition to the county-level information, the Census provides the total
population of many cities and towns in 1970, 1980 and 1990. For cities and
towns, the voting age population is estimated at the time of an election by
linearly interpolating and extrapolating the information provided by the Census
Bureau. Consider the city of Novice in Coleman county which had an election
on January 6, 1987. Novice had a voting age population of 129 in 1980 and 140
2 9 Specifically, 12 observations did not identify the precise nature of the changes proposed by
the referendum, 15 elections occurred in cities not identified in the United States Census and
37 occurred in justice precincts where the precise number of justice precincts in the county
could not be identified with confidence.
3 0 We sent letters to the clerks of the 180 counties which had liquor elections over the period
requesting information on whether other issues were being voted on at the same time. Almost
half sent copies of the notes from the Commissioners Court’s meeting or a copy of the official
document containing the results of the election. Both of these identified all items that were
voted on at the same time as the local liquor referendum. Most of the other county clerks sent
letters indicating whether the liquor law referendum was the only item on the ballot. A few
county clerks either did not respond or could not determine all items on the ballot. Of the 43
elections we suspect might have been held with other issues, 24 were ones for which we could
not get a response from the relevant county clerk and which were held on uniform election
days. The remaining 19, were ones that we knew for certain were held with other issues. Of
these, 16 were held on uniform election dates.
3 1 A jurisdiction can prohibit the retail sale of all alcohol while still allowing private clubs
(including the VFW, American Legion and other fraternal organizations) to serve alcohol.
3 2 All of these elections had at least one vote for the referendum.
27
in 1990. By linearly interpolating this information, we estimate Novice’s voting
age population to be 136.7 at the time of the election. If the election occurred in
1993, we would estimate the population by using Novice’s voting age population
in 1990 and assuming that this population grew at the same rate as the county’s
voting age population between 1990 and 1993. Because Coleman’s voting age
population grew -1.56 percent from 1990 to 1993, we would estimate the voting
age population in Novice in 1993 to be 137.8. A similar extrapolation is used for
elections prior to 1980 in cities and towns whose populations were not reported
by the 1970 Census (but were in 1980 and 1990). As with the justice precincts,
the fraction of a city’s or town’s population over the age of 50 is assumed to be
the same as in the county.
Churches & Church Membership in the United States provides county-level
information on the number of adherents to Baptist denominations. It is published every ten years (in 1970, 1980, and 1990). The total number of Baptists
in a county at the time of an election is estimated by linearly interpolating and
extrapolating this information. By dividing this number by the county population, we obtain an estimate of the fraction of the county population that is
Baptist. We use this fraction as a proxy for the fraction of Baptists in each of
the 363 jurisdictions; thereby, implicitly assuming that Baptists are uniformly
distributed throughout each county.
The United States Carbon Dioxide Information Analysis Center (CDIAC)
collects daily observations of maximum temperature, minimum temperature,
precipitation and snowfall from 1,062 weather stations (of which 44 are located
in Texas) comprising the United States Historical Climatology Network. We
calculate the midpoint of the maximum and minimum temperature at each
weather station on the day of an election and use this measure of temperature
in our specification.
9.2
Proofs
Proof of Proposition 1: Let (γ ∗s , γ ∗o ) be an interior equilibrium. Then, (γ ∗s , γ ∗o )
must satisfy the pair of first order conditions:
∂π(γ ∗s , γ ∗o )
γ∗
b= s
∂γ s
c
and
−
We know that
γ∗
∂π(γ ∗s , γ ∗o )
x = o.
∂γ o
c
γo
γo
∂π(γ s , γ o )
= h(
; ν, ω)
∂γ s
γo + γs
(γ o + γ s )2
and
−
γo
γs
∂π(γ s , γ o )
= h(
; ν, ω)
.
∂γ o
γo + γs
(γ o + γ s )2
28
It follows that the two first order conditions imply that
√
b
∗
γ s = √ γ ∗o .
x
Substituting this into the first of the two first order conditions, we find that
√ √
√
x
c( b)( x)3
∗ 2
(γ o ) = h( √
√ ; ν, ω) √
√
b+ x
( b + x)2
which implies that
(γ ∗s )2
Thus,
√
√
√
x
c( b)3 ( x)
= h( √
√ ; ν, ω) √
√ 2.
b+ x
( b + x)
γ ∗o
√ √
√
x
c( b)( x)3 1
√
√
= [h(
]2 ,
√ ; ν, ω)
√
b+ x
( b + x)2
γ ∗s
√
√
√
c( b)3 ( x) 12
x
= [h( √
] .
√ ; ν, ω) √
√
b+ x
( b + x)2
and
For the Beta distribution, we have that
√ ν−1 √ ω−1
√
x
x
b
√
,
h( √
√ ; ν, ω) = √
ν+ω−2
b+ x
( x + b)
B(ν, ω)
and substituting this into the above formulas yields the characterization stated
in the proposition. QED
Proof of Proposition 2: We need to show that γ ∗s maximizes the supporters’
γ2
payoff π(γ s , γ ∗o ; ν, ω)b − 2cs subject to the constraint that γ s ∈ [0, c] and that γ ∗o
γ2
maximizes the opposers’ payoff −π(γ ∗s , γ o ; ν, ω)x − 2co subject to the constraint
that γ o ∈ [0, c]. We prove only the former claim, since the argument for the
latter is analogous.
If γ ∗s did not maximize the supporters’ payoff, there must exist some γ
bs that
would yield a higher payoff. By condition (iii) of the Proposition, we know that
γ
bs 6= 0. Define the function ϕ : [0, c] → < as follows:
ϕ(γ s ) = π(γ s , γ ∗o ; ν, ω)b −
γ 2s
.
2c
Note first the following important claim.
γ s ) = 0 for some γ
es ∈ (0, c]. Then, ϕ00 (e
γ s ) has the
Claim: Suppose that ϕ0 (e
∗
opposite sign from (ν + 2)e
γ s − (ω − 2)γ o .
Proof: We have that
ϕ0 (γ s ) =
∂π(γ s , γ ∗o )
γ
b − s,
∂γ s
c
29
and that
ϕ00 (γ s ) =
∂ 2 π(γ s , γ ∗o )
1
b− .
2
∂γ s
c
Observe that
γo
γo
∂ 2 π(γ s , γ o )
=−
[hµ
+ 2h],
2
3
∂γ s
(γ o + γ s )
γo + γs
so that
ϕ00 (γ s ) = −
γ ∗o
1
γ ∗o
[hµ
+ 2h]b − .
∗
3
(γ s + γ o )
γ s + γ ∗o
c
For the Beta distribution, for all (γ s , γ o ) we have that
h=
and
hµ =
γ ν−1
γ ω−1
o
s
,
(γ o + γ s )ν+ω−2 B(ν, ω)
γ ω−2
γ ν−2
o
s
[(ν − 1)γ s − (ω − 1)γ o ],
(γ o + γ s )ν+ω−3 B(ν, ω)
so we may write
hµ =
[(ν − 1)γ s − (ω − 1)γ o ](γ o + γ s )h
.
γoγs
Moreover, the fact that ϕ0 (e
γ s ) = 0 implies that
It follows that
ϕ00 (e
γs) = −
1
γ ∗o
.
= hb
c
γ
es (e
γ s + γ ∗o )2
γ ∗o hb
[(ν − 1)e
γ s − (ω − 1)γ ∗o ]
2
1
{
+
+ }.
(e
γ s + γ ∗o )2
γ
es (e
γ s + γ ∗o )
(e
γ s + γ ∗o ) γ
es
Since γ ∗o > 0, the sign of ϕ00 (e
γ s ) is the opposite of the sign of
[(ν − 1)e
γ s − (ω − 1)γ ∗o ]
2
1
+
+ .
∗
∗
γ
es (e
γs + γo)
(e
γs + γo) γ
es
This is positive if (ν + 2)e
γ s > (ω − 2)γ ∗o and negative if (ν + 2)e
γ s < (ω − 2)γ ∗o .
The Claim now follows. QED
Suppose first that γ
bs > γ ∗s . Consider the problem
bs ]}
min{ϕ(γ s ) : γ s ∈ [γ ∗s , γ
Since ϕ is continuous and the constraint set is compact, the problem has a
solution which we denote by γ
es . Note that the solution must lie in the interior
of [γ ∗s , γ
bs ]. To see this note that γ
es must be less than γ
bs since ϕ(b
γ s ) > ϕ(γ ∗s ).
0 ∗
In addition, we know that by condition (i) ϕ (γ s ) = 0, and by condition (ii)
30
and the Claim, ϕ00 (γ ∗s ) < 0. This means that for γ s slightly larger than γ ∗s that
ϕ(γ s ) < ϕ(γ ∗s ). Since ϕ is smooth, it follows that ϕ0 (e
γ s ) = 0 and ϕ00 (e
γ s ) ≥ 0.
00
By the Claim, we have that ϕ (e
γ s ) ≥ 0 if and only if (ν + 2)e
γ s ≤ (ω − 2)γ ∗o .
But we know from condition (ii) and the fact that γ
bs > γ ∗s that (ν + 2)e
γs >
∗
∗
(ν + 2)γ s > (ω − 2)γ o so this is impossible. Thus, γ
bs cannot be greater than γ ∗s .
Now suppose that γ
bs < γ ∗s . Without loss of generality, we may assume that
γ
bs solves the problem:
max{ϕ(γ s ) : γ s ∈ [0, γ ∗s ]}.
Since γ
bs ∈ (0, γ ∗s ), we know that ϕ0 (b
γ s ) = 0 and ϕ00 (b
γ s ) ≤ 0. By the Claim, we
∗
know that (ν + 2)b
γ s ≥ (ω − 2)γ o . Now consider the problem
min{ϕ(γ s ) : γ s ∈ [b
γ s , γ ∗s ]}
The problem has a solution which we denote by γ
es . Note that the solution
must lie in the interior of [b
γ s , γ ∗s ]. To see this note that γ
es must be greater
than γ
bs since ϕ(b
γ s ) > ϕ(γ ∗s ). In addition, we know that by conditions (i)
and (ii) and the Claim, ϕ0 (γ ∗s ) = 0 and ϕ00 (γ ∗s ) < 0. This implies that for γ s
slightly smaller than γ ∗s that ϕ(γ s ) < ϕ(γ ∗s ). Since ϕ is smooth, it follows that
ϕ0 (e
γ s ) = 0 and ϕ00 (e
γ s ) ≥ 0. By the Claim, we have that ϕ00 (e
γ s ) ≥ 0 if and only
if (ν + 2)e
γ s ≤ (ω − 2)γ ∗o . But we know that (ν + 2)e
γ s > (ν + 2)b
γ s > (ω − 2)γ ∗o so
∗
this is impossible. Thus, γ
bs cannot be smaller than γ s . QED
9.3
9.3.1
Deriving the probability density function for (vsj , voj )
The group rule-utilitarian model
Define the functions usj : [0, 1] × < → < and uoj : [0, 1] × < → < as follows:
√
usj (µ, ε) = µKj exp ε
and
p
x
bj √
uoj (µ, ε) = (1 − µ)Kj q
exp ε.
bbj
Then, as shown in the text, we have that
vsj = usj (µj , εj ),
and
voj = uoj (µj , εj ).
In addition, we know that µj and εj are continuous random variables having
joint probability density function
hj (µj , εj ) =
ν −1
µj j
(1 − µj )ωj −1
ε2j
1
· √ exp(− ).
B(ν j , ω j )
2
2π
31
We know that vsj = usj (µj , εj ) and voj = uoj (µj , εj ) define a one-to-one
mapping from (µj , εj ) space to (vsj , voj ) space. Thus, the joint probability
density function of (vsj , voj ) is given by:
gj (vsj , voj ) = hj (µj (vsj , voj ), εj (vsj , voj )) |Ψ|
where (µj (vsj , voj ), εj (vsj , voj )) are defined implicitly by the relations vsj =
usj (µj (vsj , voj ), εj (vsj , voj )) and voj = uoj (µj (vsj , voj ), εj (vsj , voj )) and
¯
¯
¯
|Ψ| = ¯
¯
=
∂µj
∂vsj
∂εj
∂vsj
∂µj
∂voj
∂εj
∂voj
¯
¯
¯
¯
¯
∂µj ∂εj
∂µj ∂εj
−
.
∂vsj ∂voj
∂voj ∂vsj
As shown in the text,
µj (vsj , voj ) =
and
εj (vsj , voj ) = 2 ln(voj
Note that
∂µj
=
∂vsj
and
Moreover,
q
p
p
bbj + vsj x
bj ) − 2 ln(Kj x
bj )
q p
bj
voj bbj x
q
,
p
(voj bbj + vsj x
bj )2
q p
bj
−vsj bbj x
∂µj
q
=
.
p
∂voj
(voj bbj + vsj x
bj )2
∂εj
=
∂voj
voj
and
∂εj
=
∂vsj
Thus, we have:
p
vsj x
bj
q
p ,
b
voj bj + vsj x
bj
voj
q
2 bbj
q
p
bbj + vsj x
bj
p
2 x
bj
q
p
bbj + vsj x
bj
q p
bj
2 bbj x
q
.
|Ψ| =
p
(voj bbj + vsj x
bj )2
32
This implies that
µj (vsj , voj )ν j −1 (1 − µj (vsj , voj ))ωj −1
B(ν j , ω j )
q p
bbj x
2
bj
2
1
εj (vsj , voj )
q
× √ exp(−
)·
p
2
2π
(voj bbj + vsj x
bj )2
q
p ν
bj ) j (vsj )ν j −1 ( bbj )ωj (voj )ωj −1
2( x
q
=
p
(voj bbj + vsj x
bj )ν j +ωj B(ν j , ω j )
√
√
(voj bbj +vsj x
bj ) 2 2
√
] )
(ln[
Kj x
bj
1
× √ exp(−
).
2
2π
gj (vsj , voj ) =
Next observe that, after substituting in for Kj , we may write
q
q
q
p ν +ω
p
p 2
bj ) 2 ( bbj + x
bj ) j j B(ν j , ω j )cj (voj bbj + vsj x
bj )
(voj bbj + vsj x
p
q
[
] =
,
p
Kj x
bj
( x
b )ν j +2 ( bb )ωj +2
j
j
so that
q
p ν
bj ) j (vsj )ν j −1 ( bbj )ωj (voj )ωj −1
2( x
ς 2j
1
q
√
·
exp(−
gj (vsj , voj ) =
),
p
2
2π
(voj bbj + vsj x
bj )ν j +ωj B(ν j , ω j )
where
ςj
9.3.2
q
p
= (ν j + ω j ) ln( bbj + x
bj ) + ln B(ν j , ω j ) + ln cj
q
q
p
p
b
+2 ln(voj bj + vsj x
bj ) − (ν j + 2) ln( x
bj ) − (ω j + 2) ln( bbj ).
The intensity hypothesis
Define the functions usj : [0, 1] × < → < and uoj : [0, 1] × < → < as follows:
usj (µ, ε) = µTj exp ε
and
uoj (µ, ε) = (1 − µ)Tj
where Tj =
αj bbj
cj .
x
bj
exp ε.
bbj
Then, as shown in the text, we have that
vsj = usj (µj , εj ),
33
and
voj = uoj (µj , εj ).
In addition, we know that µj and εj are continuous random variables having
joint probability density function
hj (µj , εj ) =
ε2j
µν j −1 (1 − µ)ωj −1
1
· √ exp(− ).
B(ν j , ω j )
2
2π
By the same logic as for the basic model, the joint probability density function of (vsj , voj ) is given by:
gj (vsj , voj ) = hj (µj (vsj , voj ), εj (vsj , voj )) |Ψ|
where (µj (vsj , voj ), εj (vsj , voj )) are defined implicitly by the relations vsj =
usj (µj (vsj , voj ), εj (vsj , voj )) and voj = uoj (µj (vsj , voj ), εj (vsj , voj )) and
|Ψ| =
∂µj ∂εj
∂µj ∂εj
−
.
∂vsj ∂voj
∂voj ∂vsj
Solving for µj (vsj , voj ) and εj (vsj , voj ), we obtain:
µj (vsj , voj ) =
and
εj (vsj , voj ) = ln(bbj voj + x
bj vsj ) − ln Tj x
bj .
Note that
bbj x
∂µj
bj voj
=
∂vsj
(bbj voj + x
bj vsj )2
and
∂µj
bj vsj
−bbj x
=
.
∂voj
(bbj voj + x
bj vsj )2
Moreover,
bbj
∂εj
=
∂voj
(bbj voj + x
bj vsj )
and
Thus, we have:
x
bj
∂εj
=
∂vsj
(bbj voj + x
bj vsj )
|Ψ| =
This implies that
gj (vsj , voj ) =
where
x
bj vsj
,
bbj voj + x
bj vsj
bbj x
bj
(bbj voj + x
bj vsj )2
.
ς 2j
1
(b
xj )ν j (vsj )ν j −1 (bbj )ωj (voj )ωj −1
· √ exp(− ),
2
2π
(bbj voj + x
bj vsj )ν j +ωj B(ν j , ω j )
bj vsj ) − ln αj − ln x
bj − ln bbj .
ς j = ln cj + ln(bbj voj + x
34
9.3.3
The popularity hypothesis
Define the functions usj : [0, 1] × < → < and uoj : [0, 1] × < → < as follows:
usj (µ, ε) = µSj exp ε
and
uoj (µ, ε) = (1 − µ)Sj
where Sj =
α
bj
cj
·
νj
ν j +ω j .
ωj
exp ε.
νj
Then, as shown in the text, we have that
vsj = usj (µj , εj ),
and
voj = uoj (µj , εj ).
In addition, we know that µj and εj are continuous random variables having
joint probability density function
hj (µj , εj ) =
ε2j
µν j −1 (1 − µ)ωj −1
1
· √ exp(− ).
B(ν j , ω j )
2
2π
By the same logic as for the basic model, the joint probability density function of (vsj , voj ) is given by:
gj (vsj , voj ) = hj (µj (vsj , voj ), εj (vsj , voj )) |Ψ|
where (µj (vsj , voj ), εj (vsj , voj )) are defined implicitly by the relations vsj =
usj (µj (vsj , voj ), εj (vsj , voj )) and voj = uoj (µj (vsj , voj ), εj (vsj , voj )) and
|Ψ| =
∂µj ∂εj
∂µj ∂εj
−
.
∂vsj ∂voj
∂voj ∂vsj
Solving for µj (vsj , voj ) and εj (vsj , voj ), we obtain:
µj (vsj , voj ) =
ω j vsj
ν j voj + ω j vsj
and
εj (vsj , voj ) = ln(ν j voj + ω j vsj ) − ln ω j Sj .
Note that
and
∂µj
ν j ω j voj
=
∂vsj
(ν j voj + ω j vsj )2
∂µj
−ν j ω j vsj
=
.
∂voj
(ν j voj + ω j vsj )2
Moreover,
∂εj
νj
=
∂voj
(ν j voj + ω j vsj )
35
and
ωj
∂εj
=
.
∂vsj
(ν j voj + ω j vsj )
Thus, we have:
|Ψ| =
ν j ωj
.
(ν j voj + ω j vsj )2
This implies that
gj (vsj , voj ) =
ς 2j
(ω j )ν j (vsj )ν j −1 (ν j )ωj (voj )ωj −1
1
√
exp(−
·
).
(ν j voj + ω j vsj )ν j +ωj B(ν j , ω j )
2
2π
where
9.4
b j − ln ν j .
ς j = ln(ν j voj + ω j vsj ) + ln cj + ln(ν j + ω j ) − ln ω j − ln α
Solving the constrained maximization
For the group rule-utilitarian model, the problem is to maximize (1) subject
to the constraint (2) holding for each jurisdiction j. Observe that the constraints described in (2) have a nice linear structure. Moreover, we use the
same four district specific variables to explain variation in b and x and they are
all dummy variables. Thus, zbj = zxj = (1, δ 1j , δ 2j , δ 3j , δ 4j ) where δ ij ∈ {0, 1}
for i = 1, 2, 3, 4. Since there are only sixteen possible values for the vector
(δ 1j , δ 2j , δ 3j , δ 4j ), we can divide the districts into sixteen different categories
according to their (δ 1j , δ 2j , δ 3j , δ 4j ) vectors. We then need only impose the convsj
1−v
straint for the two outliers in terms of ln( 1−v
)2 and ln( vojsj )2 values in each
oj
group. This reduces the number of feasibility constraints to thirty two.
We use the following iterative procedure to solve the problem. First, we
group the districts into the sixteen categories t ∈ {1, ..., 16} according to the
values of their (δ 1j , δ 2j , δ 3j , δ 4j ) vectors. We then select from each category
vsj
the district for which ln( 1−v
)2 is maximized, denoted j(t). Next, for each
oj
t, we maximize the likelihood function imposing only the single constraint that
vsj
ln( 1−v
)2 = β b ·zbj −β x ·zxj for district j(t). We then let T (1) denote the set of
oj
v
0
sj(t )
)2 ≤
all t for which the solution satisfies the constraints that for all t0 , ln( 1−v
oj(t0 )
β b · zbj(t0 ) − β x · zxj(t0 ) and let t(1) be the t ∈ T (1) which yields the highest value
of the likelihood function.
The second step in the iteration involves considering all possible pairs (t, t0 )
such that t, t0 ∈
/ T (1) and for each of these maximize the likelihood function
vsj
imposing only the constraints that ln( 1−v
)2 = β b · zbj − β x · zxj for districts
oj
0
j(t) and j(t ). We then let T (2) denote the set of all such (t, t0 ) pairs for which
the solution satisfies all the constraints and let (t, t0 )(2) be the (t, t0 ) pair in
T (2) yielding the highest value of the likelihood function.
The third step is to consider all possible triples (t, t0 , t00 ) such that t, t0 , t00 ∈
/
T (1) and (t, t0 ), (t, t00 ), (t0 , t00 ) ∈
/ T (2) and for each of these maximize the likelivsj
hood function imposing only the constraints that ln( 1−v
)2 = β b ·zbj −β x ·zxj for
oj
36
districts j(t), j(t0 ) and j(t00 ). We then let T (3) denote the set of all such (t, t0 , t00 )
triples for which the solution satisfies all the constraints and let (t, t0 , t00 )(3) be
the (t, t0 , t00 ) triple in T (3) which yields the highest value of the likelihood function.
We continue doing this until the mth step in which it is not possible to
find any admissible m-tuples. We then compare the value of the likelihood
function associated with t(1), (t, t0 )(2), (t, t0 , t00 )(3), etc. and pick the constraint combination that results in the highest value. This solves the problem of maximizing the likelihood function subject to the feasibility constraints
vsj
ln( 1−v
)2 ≤ β b · zbj − β x · zxj for all districts j. It also turns out to satisfy the
oj
1−v
constraints that β b · zbj − β x · zxj ≤ ln( vojsj )2 for all j and hence is the solution
to the full problem.
A similar iterative procedure is use for the intensity model. This procedure
was not required for the popularity model because the coefficient estimates of
the unconstrained maximization problem satisfied all constraints.
9.5
Estimating the rule-utilitarian model
It is possible to reinterpret the variables of the environment underlying our
model in such a way as to fit a version of the rule-utilitarian model of Feddersen
and Sandroni (2002). Under this rule-utilitarian interpretation, µ would be
the fraction of the population who believed that the change would be good for
society, b would be their estimate of the average net benefit of the change, and
x would be the estimate of the average net cost of the change of those who
thought it was a bad idea. The expected payoff of those favoring the change
would be
ν γ 2s
ω γ 2o
π(γ s , γ o ; ν, ω)b − (
+
)
ν + ω 2c ν + ω 2c
and that of those opposed to the change would be
−π(γ s , γ o ; ν, ω)x − (
ν γ 2s
ω γ 2o
+
).
ν + ω 2c ν + ω 2c
The key difference is that citizens care about average voting costs rather than
just the voting costs of those on their side of the issue. Equilibrium would be
defined in the same way and conditions analagous (but different) to those presented in Proposition 1 can be obtained. Specifically, at an interior equilibrium:
√
√
c( xν)ν ( bω)ω+2 (ν + ω) 1
√
)2 ,
γ ∗s = (
√
νω( xν + bω)ν+ω B(ν, ω)
and
γ ∗o
√
√
c( xν)ν+2 ( bω)ω (ν + ω) 12
√
) .
=(
√
νω( xν + bω)ν+ω B(ν, ω)
The main difficulty in estimating the rule-utilitarian model is that it is not
clear what the parameters ν, ω, b, and x should depend upon. The difficulty is
37
that the differences in the attitudes of those on opposite sides of the issue stem
from divergent beliefs rather than differences in objectives. Without a theory as
to why people should have different beliefs, it is not clear quite what to expect
about the determinants of ν, ω, b, and x. Nontheless, it seems reasonable to
suppose that peoples’ beliefs about what is best for society should be correlated
with what is best for them. Baptists, for example, would seem more likely to say
that more liberal liquor laws were bad for society than social drinkers! Thus, as
a first cut, it seems natural to specify the same functional relationships for the
rule-utilitarian model as for the group rule-utilitarian model.
Proceeding in this way yields the results reported in Tables 9 and 10. From
Table 9 it is clear that the effects of the various variables on ν, b, and x are
very similar to those in the group rule-utilitarian model. Table 10 shows that
the average values of vj , ω j and cj are similar to those for the rule-utilitarian
model. However, the average values of bj and xj are considerably lower.
It is not possible to directly compare this model with the group rule-utilitarian
model because the interpretation of the variables are completely different. Thus,
comparing across the models we are not holding the underlying environment
fixed. Moreover, there is no obvious way of relating the variables in the two
models. However, the basic fit of the rule-utilitarian model is virtually identical
- it explains 54% of the variation in turnout for the referenda and 48% of the
variation in turnout against the referenda.
38